Polyhedron_3<Traits>::Halfedge

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CGAL::Polyhedron_3<Traits>::Halfedge

Definition

Figure 25.1 depicts the relationship between a halfedge and its incident halfedges, vertices, and facets. A halfedge is an oriented edge between two vertices. It is always paired with a halfedge pointing in the opposite direction. The opposite() member function returns this halfedge of opposite orientation. If a halfedge is incident to a facet the next() member function points to the successor halfedge around this facet. For border edges the next() member function points to the successor halfedge along the hole. For more than two border edges at a vertex, the next halfedge along a hole is not uniquely defined, but a consistent assignment of the next halfedge will be maintained in the data structure. An invariant is that successive assignments of the form h = h->next() cycle counterclockwise around the facet (or hole) and traverse all halfedges incident to this facet (or hole). A similar invariant is that successive assignments of the form h = h->next()->opposite() cycle clockwise around the vertex and traverse all halfedges incident to this vertex. Two circulators are provided for these circular orders.

Figure 25.1: The three classes Vertex, Halfedge, and Facet of the polyhedral surface. Member functions with shaded background are mandatory. The others are optionally supported.

The incidences encoded in opposite() and next() are available for each instantiation of polyhedral surfaces. The other incidences are optionally available as indicated with type tags. The prev() member function points to the preceding halfedge around the same facet. It is always available, though it might perform a search around the facet using the next() member function to find the previous halfedge if the underlying halfedge data structure does not provide an efficient prev() member function for halfedges. Handles to the incident vertex and facet are optionally stored.

The circulators are assignable to the Halfedge_handle. The circulators are bidirectional if the halfedge provided to the polyhedron with the Items template argument provides a member function prev(), otherwise they are of the forward category.

#include <CGAL/Polyhedron_3.h>

Types

Polyhedron_3<Traits>::Halfedge::Vertex
	type of incident vertices.
Polyhedron_3<Traits>::Halfedge::Facet
	type of incident facets.

Polyhedron_3<Traits>::Halfedge::Vertex_handle
	handle to vertex.
Polyhedron_3<Traits>::Halfedge::Halfedge_handle
	handle to halfedge.
Polyhedron_3<Traits>::Halfedge::Facet_handle
	handle to facet.
Polyhedron_3<Traits>::Halfedge::Halfedge_around_vertex_circulator
	circulator of halfedges around a vertex.
Polyhedron_3<Traits>::Halfedge::Halfedge_around_facet_circulator
	circulator of halfedges around a facet.

Polyhedron_3<Traits>::Halfedge::Vertex_const_handle
Polyhedron_3<Traits>::Halfedge::Halfedge_const_handle
Polyhedron_3<Traits>::Halfedge::Facet_const_handle
Polyhedron_3<Traits>::Halfedge::Halfedge_around_vertex_const_circulator
Polyhedron_3<Traits>::Halfedge::Halfedge_around_facet_const_circulator

Polyhedron_3<Traits>::Halfedge::Supports_halfedge_prev
	≡ CGAL::Tag_true or CGAL::Tag_false.
Polyhedron_3<Traits>::Halfedge::Supports_halfedge_vertex
	≡ CGAL::Tag_true or CGAL::Tag_false.
Polyhedron_3<Traits>::Halfedge::Supports_halfedge_face
	≡ CGAL::Tag_true or CGAL::Tag_false.

Creation

Polyhedron_3<Traits>::Halfedge h;
	default constructor.

Operations

Halfedge_handle	h.opposite ()
Halfedge_const_handle	h.opposite () const	the opposite halfedge.

Halfedge_handle	h.next ()
Halfedge_const_handle	h.next () const	the next halfedge around the facet.

Halfedge_handle	h.prev ()
Halfedge_const_handle	h.prev () const	the previous halfedge around the facet.

Halfedge_handle	h.next_on_vertex ()
Halfedge_const_handle	h.next_on_vertex () const	the next halfedge around the vertex (clockwise). Is equal to h.next()->opposite().

Halfedge_handle	h.prev_on_vertex ()
Halfedge_const_handle	h.prev_on_vertex () const	the previous halfedge around the vertex (counterclockwise). Is equal to h.opposite()->prev().

bool	h.is_border () const	is true if h is a border halfedge.
bool	h.is_border_edge () const	is true if h or h.opposite() is a border halfedge.

Halfedge_around_vertex_circulator	h.vertex_begin ()

Halfedge_around_vertex_const_circulator
	h.vertex_begin () const	circulator of halfedges around the vertex (clockwise).

Halfedge_around_facet_circulator	h.facet_begin ()

Halfedge_around_facet_const_circulator
	h.facet_begin () const	circulator of halfedges around the facet (counterclockwise).

std::size_t	h.vertex_degree () const	the degree of the incident vertex, i.e., number of edges emanating from this vertex.
bool	h.is_bivalent () const	returns true if the incident vertex has exactly two incident edges.
bool	h.is_trivalent () const	returns true if the incident vertex has exactly three incident edges.

std::size_t	h.facet_degree () const	the degree of the incident facet, i.e., number of edges on the boundary of this facet.
bool	h.is_triangle () const	returns true if the incident facet is a triangle.
bool	h.is_quad () const	returns true if the incident facet is a quadrilateral.

Operations available if Supports_halfedge_vertex ≡ CGAL::Tag_true

Vertex_handle	h.vertex ()
Vertex_const_handle	h.vertex () const	the incident vertex of h.

Operations available if Supports_halfedge_facet ≡ CGAL::Tag_true

Facet_handle	h.facet ()
Facet_const_handle	h.facet () const	the incident facet of h. If h is a border halfedge the result is default construction of the handle.

Implementation

The member functions prev() and prev_on_vertex() work in constant time if Supports_halfedge_prev ≡ CGAL::Tag_true. Otherwise both methods search for the previous halfedge around the incident facet.

Next: Polyhedron_3<Traits>::Vertex